Boolean Primitives as a Constructive Foundation for Arithmetic in U8: Laws, Decompositions, and Hardware Consequences

his paper advances the thesis that the standard operations of arithmetic — addition,subtraction, multiplication, and division — need not be taken as the most primitive constructivelayer of integer arithmetic. We demonstrate, over the finite block U8 = {0, 1, . . . , 255}, thatthese operations can be recovered exactly from three lower primitives: bitwise AND (∧), bitwiseOR (∨), and bitwise complement NOP(x) = 255 − x, together with a local hereditary subtractionrule. The central algebraic object is the tripartite decomposition (C, Xp, Yp), an unpublishedstructural perspective from which XOR, OR, integer sum, and integer product all emerge asexact closed-form expressions. We verify all identities exhaustively over U 28 and present necessaryfilters for perfect-square detection. The paper concludes with hardware implications: eachidentity corresponds directly to a synthesizable digital circuit, making AND, OR, and NOTthe native constructive substrate for embedded arithmetic units rather than an implementationafterthought.